Mr Spenser in the course of
his remarks regretted that so many members of the Section were in the habit
of employing the word Force in a sense too limited and definite to be of any
use in a complete theory. He had himself always been careful to preserve
that largeness of meaning which was too often lost sight of in elementary
works. This was best done by using the word sometimes in one sense and
sometimes in another, and in this way he trusted that he had made the word occupy
a sufficiently large field of thought.

James Clerk Maxwell

The concept of force
is one of the most peculiar in all of physics. It is, in one sense, the most
viscerally immediate concept in classical mechanics, and seems to serve as
the essential "agent of causality" in all interactions, and yet the
ontological status of force has always been highly suspect. We sometimes
regard force as the cause of changes in motion, and imagine that those
changes would not occur in the absence of the forces, but this causative
aspect of force is an independent assumption that does not follow from any
quantifiable definition, since we could equally well regard force as being caused
by changes in motion, or even as merely a descriptive parameter with no
independent ontological standing at all.

In addition, there is an
inherent ambiguity in the idea of changes in motion, because it isn't obvious
what constitutes unchanging (i.e., unforced) motion. Aristotle believed it was
necessary to distinguish between two fundamentally distinct kinds of motion,
which he called natural motions and violent motions. The natural motions
included the apparent movements of celestial objects, the falling of leaves
to the ground, the upward movement of flames and hot gases in the atmosphere,
or of air bubbles in water, and so on. According to Aristotle, the cause
of such motions is that all objects and substances have a natural place or
level (such as air above, water below), and they proceed in the most direct
way, along straight vertical paths, to their natural places. The motion of
the celestial bodies is circular because this is the most perfect kind of
unchanging eternal motion, whereas the necessarily transitory motions of
sublunary objects are rectilinear. It may not be too misleading to
characterize Aristotle's concept of sublunary motion as a theory of buoyancy,
since the natural place of light elements is above, and the natural place of
heavy elements is below. If an object is out of place, it naturally moves up
or down as appropriate to reach its proper place.

Aristotle has often been
criticized for saying (or seeming to say) that the speed at which an object
falls (through the air) is proportional to its weight. To the modern reader
this seems absurd, as it is contradicted by the simplest observations of
falling objects. However, it's conceivable that we misinterpret Aristotle's
meaning, partly because we're so accustomed to regarding the concept of force
as the cause of motion, rather than as an effect or concomitant
attribute of motion. If we consider the downward force (which Aristotle would
call the weight) of an object to be the force that would be required to keep
it at its current height, then the "weight" of an object really is
substantially greater the faster it falls. More strength is required to catch
a falling object than to hold the same object at rest. Some Aristotelian
scholars have speculated that this was Aristotle's actual meaning, although
his writing's on the subject are so sketchy that we can't know for certain.
In any case, it illustrates that the concept and significance of force in a
physical theory is often murky, and it also shows how thoroughly our
understanding of physical phenomena is shaped by the distinction between
forces (such as gravity) that we consider to be causes of motion, and
those (such as impact forces) that we consider to be caused by motion.

Aristotle also held that
the speed of motion was not only proportional to the "weight"
(whatever that means) but inversely proportional to the resistance of the
medium. Thus his proposed law of motion could be expressed roughly as V =
W/R, and he used this to argue against the possibility of empty space, i.e.,
regions in which R = 0, because the velocity of any object in such a region
would be infinite. This doesn't seem like a very compelling argument, since
we could easily counter that the putative vacuum would not be the natural
place of any object, so it would have no "weight" in that direction
either. Nevertheless, perhaps to avoid wrestling with the mysterious fraction
0/0, Aristotle surrounded the four sublunary elements of Earth, Water, Air,
and Fire with a fifth element (quintessence), the lightest of all, called
aether. This aether filled the super-lunary region, ensuring that we would
never need to divide by zero.

In addition to natural
motions, Aristotle also considered violent motions, which were any motions
resulting from acts of volition of living beings. Although his writings are
somewhat obscure and inconsistent in this area, it seems that he believed
such beings were capable of self-motion, i.e., of initiating motion in the
first instance, without having been compelled to motion by some external
agent. Such self-movers are capable of inducing composite motions in other
objects, such as when we skip a stone on the surface of a pond. The stone's
motion is compounded of a violent component imparted by our hand, and the natural
component of motion compelling it toward its natural place (below the air and
water). However, as always, we must be careful not to assume that this motion
is to be interpreted as the causative result of the composition of two
different kinds of forces. It was, for Aristotle, simply the kinematic
composition of two different kinds of motion.

The bifurcation of motion
into two fundamentally different types, one for natural motions of non-living
objects and another for acts of human volition – and the attention that
Aristotle gave to the question of unmoved movers, etc. – is obviously related
to the issue of free will, and demonstrates the strong tendency of scientists
in all ages to exempt human behavior from the natural laws of physics, and to
regard motions resulting from human actions as original, in the sense
that they need not be attributed to other motions. We'll see in Section 9
that Aristotle's distinction between natural and violent motions plays a key
role in the analysis of certain puzzling aspects of quantum theory.

We can also see that the
ontological status of "force" in Aristotle's physics is ambiguous.
In some circumstances it seems to be more an attribute of motion rather than
a cause of motion. Even if we consider the quantitative physics of Galileo, Newton, and
beyond, it remains true that "force", while playing a central role
in the formulation, serves mainly as an intermediate quantity in the
calculations. In fact, the concept of 'force' could almost be
eliminated entirely from classical mechanics. (See section 4 for further
discussion of this.) Newton wrestled with the question of whether force should
be regarded as an observable or simply a relation between observables.
Interestingly, Ernst Mach regarded the third law as Newton's
most important contribution to mechanics, even though other's have criticized
it as being more a definition than a law.

Newton’s struggle to find
the "right" axiomatization of mechanics can be seen by reading the
preliminary works he wrote leading up to The Principia, such as "De motu
corporum in gyrum" (On the motion of bodies in an orbit). At one point
he conceived of a system with five Laws of Motion, but what finally
appeared in Principia were eight Definitions followed by three Laws. He
defined the "quantity of matter" as the measure arising conjointly
from the density and the volume. In his critical review of Newtonian
mechanics, Mach remarked that this definition is patently circular, noting
that "density" is nothing but the quantity of matter per volume.
However, all definitions ultimately rely on undefined (irreducible) terms, so
perhaps Newton was entitled to take density and volume as two such
elements of his axiomatization. Furthermore, by basing the quantity of matter
on explicitly finite density and volume, Newton deftly
precluded point-like objects with finite quantities of matter, which would
imply the existence of infinite forces and infinite potential energy
according to his proposed inverse-square law of gravity.

The next basic definition
in Principia is of the "quantity of motion", defined as the measure
arising conjointly from the velocity and the quantity of matter. Here we see
that "velocity" is taken as another irreducible element, like
density and volume. Thus, Newton's ontology consists of one irreducible entity,
called matter, possessing three primitive attributes, called density,
volume, and velocity, and in these terms he defines two
secondary attributes, the "quantity of matter" (which we call
"mass") as the product of density and volume, and the "quantity
of motion" (which we call "momentum") as the product of velocity
and mass, meaning it is the product of velocity, density, and volume. Although
the term "quantity of motion" suggests a scalar, we know that
velocity is a vector, (i.e., it has a magnitude and a direction), so it's
clear that momentum as Newton defined it is also is a vector. After going on to
define various kinds of forces and the attributes of those forces, Newton then,
as we saw in Section 1.3, took the law of inertia and relativity as his First
Law of Motion, just as Descartes and Huygens had done. Following this we have
the "force law", i.e., Newton's Second Law of Motion:

The change of motion is
proportional to the motive force impressed; and is made in the direction of
the right line in which the force is impressed.

Notice that this statement
doesn't agree precisely with either of the two forms in which the Second Law
is commonly given today, namely, as F = dp/dt or F = ma. The former is
perhaps closer to Newton's actual statement, since he expressed the law in
terms of momentum rather than acceleration, but he didn't refer to the rate
of change of momentum. No time parameter appears in the statement at
all. This is symptomatic of a lack of clarity (as in Aristotle’s writings)
over the distinction between "impulse force" and "continuous
force". Recall that our speculative interpretation of Aristotle's
downward "weight" was based on the idea that he actually had in
mind something like the impulse force that would be exerted by the object if
it were abruptly brought to a halt. Newton's Second Law, as expressed in the
Principia, seems to refer to such an impulse, and this is how Newton used it
in the first few Propositions, but he soon began to invoke the Second Law
with respect to continuous forces of finite magnitude applied over a finite
length of time – more in keeping with a continuous force of gravity, for
example. This shows that even in the final version of the axioms and
definitions laid down by Newton, he did not completely succeed in clearly
delineating the concept of force that he had in mind. Of course, in each of
his applications of the Second Law, Newton made the necessary dimensional adjustments to
appropriately account for the temporal aspect that was missing from the
statement of the Law itself, but this was done ad hoc, with no clear
explanation. (His ability to reliably incorporate these factors in each
context testifies to his solid grasp of the new dynamics, despite the
imperfections of his formal articulation of it.) Subsequent physicists
clarified the quantitative meaning of Newton’s second law, explicitly recognizing the
significance of time, by expressing the law either in the form F = d(mv)/dt
or else in what they thought was the equivalent form F = m(dv/dt). Of course,
in the context of special relativity these two are not equivalent, and only
the former leads to a coherent formulation of mechanics. (It’s also worth
noting that, in the context of special relativity, the concept of force is
largely an anachronism, and it is introduced mainly for the purpose of
relating relativistic descriptions to their classical counterparts.)

The third Law of Motion in
the Principia is regarded by many people as one of Newton's
greatest and most original contributions to physics. This law states that

To every action there is always
opposed an equal reaction: or, the mutual actions of two bodies upon each
other are always equal, and directed to contrary parts.

Unfortunately the word
"action" is not found among the previously defined terms, but in
the subsequent text Newton clarifies the intended meaning. He says "If a
body impinge upon another, and by its force change the motion of the other,
that body also... will undergo an equal change in its own motion towards the
contrary part." In other words, the net change in the "quantity of
motion" (i.e., the sum of the momentum vectors) is zero, so momentum is
conserved. More subtly, Newton observes that "If a horse draws a stone tied
to a rope, the horse will be equally drawn back towards the stone". This
is true even if neither the horse nor the stone are moving (which of course
implies that they are each subject to other forces as well, tending to hold
them in place). The illustrates how the concept of force enables us to
conceptually decompose a null net force into non-null components, each
representing the contributions of different physical interactions.

In retrospect we can see
that Newton's three "laws of motion" actually represent
the definition of an inertial coordinate system. For example, the
first law imposes the requirement that the spatial coordinates of any
material object free of external forces are linear functions of the time
coordinate, which is to say, free objects move with a uniform speed in a
straight line with respect to an inertial coordinate system. Rather than
seeing this as a law governing the motions of free objects with respect to a
given system of coordinates, it is more correct to regard it as defining a
class of coordinates systems in terms of which a recognizable class of
motions have particularly simple descriptions. It is then an empirical
question as to whether the phenomena of nature possess the attributes
necessary for such coordinate systems to exist.

The significance of “force”
was already obscure in Newton’s three laws of mechanics, but it became even more
obscure when he proposed the law of universal gravitation, according to which
every particle of matter exerts a force of attraction on every other
particle of matter, with a strength proportional to its mass and inversely
proportional to the square of the distance. The rival Cartesians expected all
forces to be the result of local contact between bodies, as when two objects
press directly against each other, but Newton’s conception of instantaneous gravity between
distant objects seems to defy representation in those terms. In an effort to
reconcile universal gravitation with semi-Cartesian ideas of force, Newton’s
young friend Nicolas Fatio hypothesized an omni-directional flux of small “ultra-mundane”
particles, and argued that the mutual shadowing effect could explain why
massive bodies are forced together. The same idea was later taken up by
Lesage, but many inconsistencies were pointed out, making it clear that no
such theory could accurately account for the phenomena. The simple notion of
force at a distance was so successful that it became the model for all mutual
forces between objects, and the early theories of electricity and magnetism
were expressed in those terms. However, reservations about the
intelligibility of instantaneous action at a distance remained. Eventually Faraday
and Maxwell introduced the concept of disembodied “lines of force”, which
later came to be regarded as fields of force, almost as if force was
an entity in its own right, capable of flowing from place to place. In this
way the Maxwellians (perhaps inadvertently) restored the Cartesian ideas that
all space must be occupied and that all forces must be due to direct local contact.
They accomplished this by positing a new class of entity, namely the field.
Admittedly our knowledge of the electromagnetic field is only inferred from
the behavior of matter, but it was argued that explanations in terms of
fields are more intelligible than explanations in terms of instantaneous forces
at a distance, mainly because fields were considered necessary for strict
conservation of energy and momentum once it was recognized that
electromagnetic effects propagate at a finite speed.

However, the explanation of
phenomena in terms of fields, characterized by partial differential
equations, was incomplete, because it was not possible to represent stable
configurations of matter in these terms. Maxwell’s field equations are
linear, so there was no hope of them possessing solutions corresponding to
discrete electrical charges or particles of matter. Hence it was still
necessary to retain the laws of mechanics of discrete entities, characterized
by total differential equations. The conceptual dichotomy between Newton’s
physics of particles and Maxwell’s physics of fields is clearly shown by the
contrast between total and partial differential equations, and this contrast
was seen (by some people at least) as evidence of a fundamental flaw. In a
1936 retrospective essay Einstein wrote

This is the basis on which H.
A. Lorentz obtained his synthesis of Newton’s mechanics and Maxwell’s field
theory. The weakness of this theory lies in the fact that it tried to
determine the phenomena by a combination of partial differential equations
(Maxwell’s field equations for empty space) and total differential equations
(equations of motions of points), which procedure was obviously unnatural.

The difference between
total and partial differential equations is actually more profound than it
may appear at first glance, because (as alluded to in section 1.1) it entails
different assumptions about the existence of free-will and acts of volition.
If we consider a point-like particle whose spatial position x(t) is strictly
a function of time, and we likewise consider the forces F(t) to which this
particle is subjected as strictly a function of time, then the behavior of
this particle can be expressed in the form of total differential equations,
because there is just a single independent variable, namely the time
coordinate. Every physically meaningful variable exists as one of a countable
number of explicit functions of time, and each of the values is realized at it’s
respective time. Thus the total derivatives are evaluated over actualized
values of the variables. In contrast, the partial derivatives over immaterial
fields are inherently hypothetical, because they represent the variations in
some variable of a particle not as a function of time along the particle’s
actual path, but transversely to the particle’s path. For example, rather
than asking how the force experienced by a particle changes over time, we ask
how the force would change if at this instant of time the
particle was in a slightly different position. Such hypotheticals have
meaning only assuming an element of contingency in events, i.e., only if we
assume the paths of material objects could be different than they are.

Of course, if we were to
postulate a substantial continuous field, we could have non-hypothetical
partial derivatives, which would simply express the facts implicit in the
total derivatives for each substantial part of the field. However, the
intelligibility of a truly continuous extended substance is questionable, and
we know of no examples of such a thing in nature. Given that the elementary
force fields envisaged by the Maxwellians were eventually concede to be
immaterial, and their properties could only be inferred from the state
variables of material entities, it’s clear that the partial derivatives over
the field variables are not only hypothetical, but entail the assumption of
freedom of action. In the absence of freedom, any hypothetical transverse
variations in a field (i.e., transverse to the actual paths of material
entities) would be meaningless. Only actual variations in the state variables
of material entities would have meaning. Thus the contrast between total and
partial differential equations reflects two fundamentally different
conceptual frameworks, the former based on determinism and the latter based
on the possibility of free acts. This is closely analogous to Aristotle’s
dichotomy between natural and violent motions.

As noted above, Einstein
regarded this dualism as unnatural, and his intuition led him to expect that
the field concept, governed by partial differential equations, would
ultimately prove to be sufficient for a complete description of phenomena. In
the same essay mentioned above he wrote

What appears certain to me,
however, is that, in the foundations of any consistent field theory, there
should not be, in addition to the concept of the field, any concept
concerning particles. The whole theory must be based solely on partial
differential equations and their singularity-free solutions.

It may seem ironic that he
took this view, considering that Einstein was such a staunch defender of
strict causality and determinism, but by this time he was wholly committed to
the concept of a continuous field as the ultimate ontological entity, more
fundamental even than matter, and possessing a kind of relativistic substantiality,
subject to deterministic laws. In a sense, he seems to have come to believe
that the field was not a hypothetical entity inferred from the observed behavior
of material bodies, but rather that material bodies were hypothetical
entities inferred from the observed behavior of fields. An important first
step in this program was to eliminate the concept of forces acting between
bodies, and to replace this with a field-theoretic model. He (arguably)
accomplished this for gravitation with the general theory of relativity,
which completely dispenses with the concept of a "force of
gravity", and instead interprets objects under the influence of gravity
as simply proceeding, unforced, along the most natural (geodesic) paths. Thus
the concept of force, and particularly gravitational force, which was so
central to Newton's synthesis, was simply discarded as having no
absolute significance.

However, the concept of
force is still very important in physics, partly because we continue to
employ the classical formulation of mechanics in the limit of low speeds and
weak gravity, but more importantly because it has not proven possible
(despite the best efforts of Einstein and others) to do for the other forces
of nature what general relativity did for gravity, i.e., to express the apparently
forced (violent) motions as natural paths through a modified geometry of
space and time.